Ritz method

They solved these equations using a finite element method (Ritz-Galerkin... 

MacDonald J K L 1933 Successive approximations by the Rayleigh-Ritz variation method Phys. Rev4Z 830-3... 

Klahn, B. and Bingel, W. A. [19 77] The Convergence of the Rayleigh-Ritz Method in Quantum Chemistry , Theoretica Chimica Acta, 44, p. 9. 

Schwartz, H. M., Phys. Rev. 103, 110, "Ground state solution of the non-relativistic equation for He." More rapid convergence in the Ritz variational method by inclusion of half-integral powers in the Hylleraas function. 

A treatment of the hydrogen molecule by the Ritz method, applied to helium by Kellner (25), has been reported by S. C. Wang (Phys. Rev., 31, 579 (1928)). With this method the individual eigenfunctions p and

hydrogen-like eigenfunctions of an atom with atomic number Z differing from unity. The value found for Z is 1.166, and the... 

B. N. Finkelstein and G. E. Horowitz (Z. f. Physik, 48, 118 (1928)) have similarly applied the Ritz method to the hydrogen molecule-ion, obtaining the following values ... 

Variational difference methods (the Ritz method and the Bubnov-Ga-lerkin method). The Ritz and the Bubnov-Galerkin variational methods have had considerable impact on complex numerical modeling problems and designs of difference schemes. 

The main idea behind the Ritz method is to take into consideration... 

So, the three-point scheme (30) (32) constructed by the Ritz method is identical with scheme (12) obtained by means of the IIM. In contrast to the Ritz method the Bubnov-Galerkin method applies equally well to... 

For r[x) = 0 this scheme is identical with scheme (31)-(32) obtained by means of the Ritz method. In the case of constant coefficients k x), r(x)... 

When the coordinate functions y>iix) = y x — x )/h) are chosen by an approved rule as suggested before, the Ritz and the Bubnov-Galerkin methods coincide with the finite element method. 

The finite-element method (FEM) is based on shape functions which are defined in each grid cell. The imknown fimction O is locally expanded in a basis of shape fimctions, which are usually polynomials. The expansion coefficients are determined by a Ritz-Galerkin variational principle [80], which means that the solution corresponds to the minimization of a functional form depending on the degrees of freedom of the system. Hence the FEM has certain optimality properties, but is not necessarily a conservative method. The FEM is ideally suited for complex grid geometries, and the approximation order can easily be increased, for example by extending the set of shape fimctions. 

But does all this biodiversity have any consequences for soil processes at the macro-scale This is a seemingly straightforward question, but the answer has been surprisingly elusive. Progress has been hampered by the absence of suitable experimental methods for analysing biological diversity and its relation to soil fnnctions. Three types of method are nsed (Ritz, 2004) ... 

Milde, P., Merke, J., Ritz, E., Haussler, M., and Rauterberg, E. (1989) Immuno-histochemical detection of 1,25-dihydroxyvitamin D3 receptors and estrogen receptors by monoclonal antibodies comparison of four immunoperoxidase methods. J. Histochem. Cytochem. 37,1609-1617. 

Abstract. An application of the Rayleigh-Ritz variational method to solving the Dirac-Coulomb equation, although resulted in many successful implementations, is far from being trivial and there are still many unresolved questions. Usually, the variational principle is applied to this equation in the standard, Dirac-Pauli, representation. All observables derived from the Dirac equation are invariant with respect to the choice of the representation (i.e. to a similarity transformation in the four-dimensional spinor space). However, in order to control the behavior of the variational energy, the trial functions are subjected to several conditions, as for example the kinetic balance condition. These conditions are usually representation-dependent. The aim of this work is an analysis of some consequences of this dependence. 

The main idea behind the Ritz method is to take into consideration a sequence of finite-dimensional spaces Vn with basis functions (p, i =... 

The Ritz method proves to be useful in studying the problem... 

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